Canadlan MineraloglstVol. 14, pp.334-345 (1976)

THE CRYSTAT GHEMISTRY OF THE AMPHIBOLES: l\f. X'RAY AND NEUTRONREFINEMENTS OF THE GRYSTAL STRUCTURE OF TREMOTITE

F. C. HAWTHORNE1 e'ro H. D. GRUNDYDept. of Geology, McMaster University, Hamilton, Ontorlo, Canada

ABSTRACT

Tlrroe-dimensional counter-collected X-ray andneutron single-crystal data have been used to refinethe crystal structure of a tremoilite (N%.uotrG.usCar.sorxMgsxAl6.e2ssi?.,67)O,2(OHr.$rFo.o"oClo.orz); a9.863(1) b t8.O48Q), c 5.285(1)A, p 1,04.79(r)",Z:2 in, the space group C2/m. The final R-factorsof. 3.2Vo (X-ray) and 2.lVo (neutron) were obtainedtor 1376 and 709 observed reflections respectively.

The neutron study confirmed the position of thehydrogen atom proposed by Papike et al. (1969);the O(3)-H distance of 0.960(6)A is typical for hy-droxyl anion and the principal stretching frequencyof 3684cm{ indicates little or no hydrogen bondingcrith the chain-bridsing anions. Comparison of thepositional parameters obtained from both refrne-ments shows that the positional parameters obtainedin tbo X-ray study are not significantly affestedby bonding effects. On the otler han4 differencesdo occur between the corresponding thermal para-met€rs; the X-ray R.M.S. thermal displacements aresystematically larger than the neutron drsplacements,indicating a significant contribution from electrondelocalization. In both studies, the amount of vlbra-tional anisotropy fu related to the coordination num-ber. The orientation of the vibration ellipsoids isrelated to the disposition of the surrounding atoms;the minimum vibration direction is gBnerally sub-parallel to the strongest bond, rvhereas the maximumvibration direction is generally oriented so as toinvolve the minimum bond stretching.

SoMvrens

Les donn6es recueillies sur monocristal par dif-fraction de rayons-X et d9 neutrons avec comlFteurs tridimensionnels ont 6t6 utilisdes pour affinerla structure cristalline de tr6molite Na".a8rKo.r,.,Car.sor)(Mgs)(Alo.ezeSiz.zoz)Oz:(OHr.rrrFo.uuoClo.lr); o9.863(1), b 18.048(2), c 5.285(1)4, p 104.79(t)"Z:2 dans le groupe spatial C2/m. I-es derniersindices R de 3.2Vo pour 1es rayons-X et 2J% rrcvrles neutrons ont 6t6 obtenus pour 1396 et 709 16-floxions observdes, respectivement. L6tude par neu-trons a confirm6 la position de I'atome d'hydrogbne,propos6e pa1 Papike et al. (1969); la distance O(3)-IIde 0.960(6)A est typique pour l'anion hydroxyl et

rPresent address: Dept. of Earth Sciences, Llniver-sity of Manitoba, Winnipeg, Manitoba, Canaila,R3T 2N2.

la priucipale frdquence d'allongement de 3684 cmiindique essentiellement aucun lien de l'hydrogdneavec les anions qui lient les chatnes. Une compa-raison entre les coordono6es de position atomiqueobtenues dans les deux affinements d6montre queles parambtres de position obtenus par rayons-Xno sont pas affect6s de fagon significative par lesoffets de lien. Par contre, il y a des diff6rencesentre les parambtres thermiques correspondants; lesd6placements thermiques R.M.S. d6termin6s parrayon-X sont syst6matiquement plus grands queceux d6termin6s par neutrons, ce qui indique unecontribution provenant de la d6localisation des 6lec-trons. Dans los deux 6tudes, le degr6 d'anisotropievibrationnelle est fonction de la coordination desatomes. Lbrientation des efipsoides de vibration esten rapport avec la disposition des atomes environ-nants; Ia direction de la vibration minimum e$ 96-n6ralement sub-parallDle au lien le plus fort, taodisque la drrection de vibration maximum est g6n6ra-lement orient6e de facon i entralner un allongementminimum des liens.

Clraduit par le journal)

IurnooucrroN

The past few years have seen an increase inthe number of X-ray structure refinements ofminerals at high temperatures, and considerableattention has been focused on the behavior ofthe atomic thermal paramete$. It h well knowntlat the anisotropic thermal parameters derivedfrom X-ray data often absorb deviations fromspherical symmetry of the atomic charge sloud,and thus temperature factors obtained in thisway may be systematically in error due to bond-ing effects. This effect is not present when thediffraction experiment is performed with neu-trons as an atom may be accurately representedby a point scatterer in this case. Thus a com-parison of the results of these two experimentsgives an indication of the systematic error in-herent in the derived X-ray model. This hasbeen shown to be considerable for atoms fromthe first row of the periodic table (Coppens1974), to tle extent that the positional para-meters are also significantly affected. In orderto assess this effect for (geologically) morerelevant atoms, the crystal structure of a sligftflyedenitic tremolite has been investigated by bothmethods. This also has the additional advantage

334

CRYSTAL STRUCTURE OF TREMOLITE

TAELE 1. CRYSIAI. DATA & DATA COLIiECTION IMOR!,TATION FOR TREMOLITE

335

Cheqlel Analyslg utrtt cell coutents! Mlscellatreua

si02

T102

Alzo3

F"203

Fe0

l'ftr0

Mgo

UEU

Naro

*20

'zo

56.57

0 .01

1.4t"

0 . 0 1

0 .08

0 .03

24.4L

1 t t <

L . 4 4

0 . 68

r . 4 6

0 .0s

r . 5 2

si

Tetrahedral

elvr

l a

t e

! e

Mn

Mg

Octahedral

Na

K

Large Catlon

OH

F

7 . 7 ( ' 7 a E 9 . S 6 3 ( 1 ) g

0 . 2 2 8 b - 1 8 . 0 4 8 ( 2 ) t

L 2 2 2 c E s . 2 8 5 ( r ) t

P - L04.79(L)o

0.001 x_Ray NeutrouLlnear absorptloll

0.001. coefficLent (cn-r) L4.3 0.004

0.009 Crysral stze (m) 0.05x0.15x0.25 5x7x8

0.003 Radlatlon/Monochronator t,o/C

4.995 crystal- Axls for 10idata collectlon

5 . 0 0 9No. of non-equivalent

1 . 8 0 2 l F . | > 4 d L 3 7 6 7 0 9I o o s t .

0 .383 F ina l R (obsened) 3 '22 z .LZ

0.119 F lna l R (a11 re f lec t lons) 4 .47" 2 ,32

2.304 Standard deviatLoD of unttwe lgh t obaerva t lon l - .83 .O.22

1 ' 337 Ex t lnc t lon coef f l c l .en t 1 .4 (1)x10-5

0.660 F lna l R (observed) 3 ,5%w

0.012 F lna l R (a11 re f l "ec t lons) 4 .67 .

Space croup

DeDsity -- carc

z

e2ln

2.964

2

3 .2 ( 2 ) x1o -4

t 1 d /

3 . 2 2

Temperalure factor

forn ueed:

R =x ( Iroo"l - l.""r"l )/ f, lr,o"l

n, - ( X w( | ron" l - l , , ,^ r "1> ' f E"Foo"2)k, . " = 1

l- Cs lcu la t lon baaed on 24(0 ,0H,F,C1, ) .

of providing accurate positional parameters forthe hydrogen atom adjacent to O(3) in thestructure.

The crystal structure of tremolite \ilas de-termined by Warren (1929) and. a twodimen-sional refinement was presented by Zussman(1959). A fulI three-dimensional X-ray refine-ment was performed by Papike et al. (L969) ona fairly pure tremolite from the Gouverneurdistrict of New York; this work approximatelylocated the position of the hydrogen atom inthe strucfure.

EIrPERI\4BNTAL

The crystal used in this study was from tleMcMaster University mineral collection, andwas odginally collected from the Gouverneurdistrict, New York, although the exact locationis unknown. The original sgmple was a largesingle crystal measuring 10X4X2cm, elongatealong Z and with the form {110}. Opticalexamination of both polished and thin sectionsshowed it to be free of inclusions and apparent-ly unzoned.

All single-crystal X-ray precession photo-graphs of small fragments displayed diffractionsymmetry 2/ rnC-/ - consistent with space groupsC2, Crn and CZ/m. Following previous work(Zussman 1959; Papike et aI. 1969), the spacegroup C2/ m was assigned. Unit-cell parameterswere determined from fifteen intense reflectionsaligned automatically on a four-circle diffrac-tometer; the values obtained from a least-squares refinement (fable 1) were confirmedby repeating this procedure on several differentcrystals. Subsequent to the collection of tleneufton data, the crystal used for the collectionof neufon intensities was broken in half,crushed and analyzed Clable 1).

Neutron study

An equidimensional crystal of volume -03cme was cut from the centre of the large crystalThis was mounted with epoxy on an aluminiumpin and aligned by X-ray precession photographysuch that theTOt direction was coincident withthe spindle axis. Intensity data were collected

on a semi-automatic four-circlb diffractometerrin the McMastef University nuclear teactor.The incident neutron beam was obtained ftomthe Q2O) face of a single crystal of aluminiumin the transmission geometry. Because of themosaic spread in the aluminiun crystal, thewavelength of the incident beam was variableacross its width. The wavelength was measutedat the sample crystal position during the orienta-tion procedure; the value obtained" 1.05926jt"indicates that secondary contamination is verysmall. The crystal was aligned manually, andseveral peak profiles were examined to ensurethat the mosaic spread of the sample crystalwas not excessive. Diffracted intensities weremeasured with a helium-3 counter usingar-sc&DS and operating in the step-scan mode.The scan range was a step-function of two-thetaand was wide enough to include several back-ground counts on either side of the peak. Tonegate the effect of flux variations, countingtime was controlled by a fixed-count monitoron the incident beam. Diffrastometer settingswere calculated using ttre program MACDIF,written by Martin Anderson and modified byFrank Hawthorne. Data collection was con-trolled by paper tape input through an on-lineteletype. Reflections were collected up to lfi)"Za (sin 0/X= 0.723) over one asymmetric unit.Due to the low intensity of the incident beam,the rate of data collection was slow (-10 reflec-tions per day). To avoid collecting many unob-served reflections, approximate neutron in-tensities were calculated from the atomicparameters of preliminary X-ray refinement le-sults (R-6%), and the neutron reflections werecollected in sets of approximately equal in-tensrty. As the intensities of the collected reflec-tions desreased, the incident beam monitor countinterval was increased in order to maintain goodcounting statistics. One standard reflection wassaamined each day (- once per 10 reflections)to ensure constancy of crystal alignment. Thepeak profile of each reflection was examinedand those showing any irregularity wete re-collected.

A total of 73t unique reflections was col-lected, corrected for background and the Lorentzeffect and reduced using the program DIFDAT,written by Martin Anderson and modified byFrank Hawthorne. The resulting structure fac-tors were classed as unobserved if their mag-nitude fell below that of four times the standarddeviation based on counting statistics; this re-

336

lBuilt by Martin Anderson,McMaster Universtty.

TIIB CANADIAN MINBRALOGIST

sulted in 709 observed reflections. Because ofthe extremely small absorption coefficient forneutrons, no absorption corrections were per-formed.

X-ray study

A regular cleavage fragment of dimensions0.5X0.15X0.26 mm was used to collect theintensity data. The crystal was mounted on a

Syntex Pi automatis four-circle diffractometeroperating n the 0-20 scan mode with variablescan rates fuom 2.V24.0o/min.o depending onthe peak count through an angle of 2o andthe ira, separation. Graphite-monochromatedMoKo radiation (I-0.71069A) was used, andbackground counts were made at the beginningand end of each scan. TWo standard reflectionswere monitored every 50 reflections; no signi-ficant change in their intensities was obsenedduring the data collection. A total of 3191 re-flections ltras mquuled over two asymmetricunits out to a20 value of 65o (sin 0/\ - A.754).The data were corrected for absorption, usingan eight-point gaussian quadrature integrationfor polyhedral crystal shape, for background'Lorentz and polarization effects. Equivalentreflections were averaged to produce an aslm-metric set and converted to structure factors.The resulting lFo"l were classed as observed iftheir magnitude exceeded that of fow standarddeviations based on counting statistics. Of the1640 unique reflections produced, 1376 wereclassed as observed.

RernrElv{sl{r

Crystal-structure refinement was carried outusing the least-squares program RFINE (Finger1969). Quoted R-factors are of the forms givenin Table 1 and are expressed as percentages;anisotropic temperature factors are of the formgiven in Table 1.

NeatronCoherent neutron scattering amplitudes were

taken from Bacon (1972). Atomic coordinatesand isotropic temperature factors for tremolite(Papike et al. 1969) were used as the initialmodel. One cycle of least-squares refinementvarying the atomic positions resulted in an R-factor of II.47o, and a further cycle varyingthe isotropic temperature factors gave an R-factor of 4.1/o. An additional cycle of refine'ment varying positions and isotropic temperaturefactors converged to an R-value of 3.87o.

Difference fourier sections in the vicinity ofthe u4-site revealed a splitting of the alkali atoms@apike et al. 1969; Hawthorne & Grundy1972, l973ub); using the split ./-site positions

Dept. of Physics,

^ / r \ 0 . 1 1 1 2 ( 2 )0 . 1 1 1 4 ( r )

^ / ? \ 0 . 1 1 8 9 ( 2 )0 . 1187 ( r )

^ / 1 , 0 . 1 0 7 2 ( l )0 .1082(2)o.3646(2'o . 3 6 4 2 ( 2 \

^ /< \ 0 .3466(2 '0 .31 ,67 ( r ,

^ /4 \ 0 .3437(2)0 . 3 4 3 7 ( 1 )0 .3386 (3 )0 . 3 3 8 0 ( 2 )

r / 1 \ 0 . 2 8 0 0 5 ( 7 )0 .2799 (2 )

r ( 1 \ 0 . 2 8 8 0 3 ( 7 )o.2882(2)

M(1) 3l1(2) 3r{(3) 3M(4) 3

0 . 0 4 0 7 ( 2 3 )0 . 0 4 5 0 ( 4 0 )

A(2) 3u i.zoaarol

0.0858(1) 0 .2184(3) 0 .52(2)o .o857o(7) 0 .2182(3) 0 .44(2)0 .1706(1) 0 .7241 (3 ) 0 ,5s(2)0 .17067(6) 0 .7251(3) O.46(2)0 0 . 7 1 5 2 ( 5 ) 0 . 7 6 ( 3 )o 0 .1L54(4 ' ) 0 .62(3)0 .2482(1) 0 .7931(3) 0 .70(2)0 .24818(7) O.7928(3) 0 .61(2)0 .1341(1) 0 .1003(3) 0 .77(3)0 .1339r (7) 0 .0998(3) 0 .64(2)0 .1183(1) 0 .5913(3) 0 .69(2)0 .11805(7) 0 ,5910(3) 0 .62(2)0 0 .2913 (5 ) 0 .76 (4 )0 o .292rG) 0 .7s(3)0 .08417(3) 0 .2979( r ) 0 .43(1)0.6424(4) 0.2974(4) 0.32(2)0 .17132(4) 0 .8050(1) 0 .42(1)0 .17133(8) 0 .80s6(4) 0 .37(3)0 .08821(7) L /2 0 .54(2)0 .0883(1) r /2 0 .37(3)0 .17678 (6 ) 0 0 .51 (2 )0 . 1 7 7 0 ( 1 ) o 0 . 3 9 ( 3 )0 o 0 . 5 3 ( 2 )0 0 0 .35(4)0 .27797 (4 ) r /2 0 .72(2)0 .2 '179( r ) r /2 0 .72(1)u2 0 .0887(42) 2 ,1 (3)1" /2 0 .1030(76) L .4<4)0 .4901(19) 0 2 .1 (3)0 .4897(38) o 1 .4 (4)

6 i . ru r r t o> -

At@s X-Ray Neutron

r (1 ) -0 (1) 1 .6u(2)8r (1 ) -0 (5) 7 ,637 (2 )r (1 ) -0 (6) r ,639(2)r (1 ) -o (7) L .629(L)

337

X-Ray Neutron

r .613(2)8 r .617(2)81 .589(2) 1 . 586(2)r.662(2) r.658(2)r .67s(2) L ,68r (2 )

!:q9:-- l.-Ql-

2 .O7r (2> 2 .010(L)2.047(2) 2.054(2)

2.4L5(2r 2,41:Q)2.336<2) 2.337 (2)2 .766(2) 2 ,77o(2)2.548(2) 2.552<2)

2 .5L6 2 .518

3.40(1) . 3 .40(3)3 . 5 0 ( 1 ) . 3 . 4 8 ( 2 )3 .350(1) 3 .351(2)3 .728( r ) 3 ,721(2)

3 ,184(2) 3 .186(4)3 .088(1) 2 .090(2)3 .08s(x ) 3 .086(1)3.425(r) 3.422(3)3 .191(X) 3 .195(2)3 ,121(1) 3 .209(2)

0 .960(6)

CRYSTAL STRUCTURB

TABLE 2. ATO}IIC PARAXETERS !'OR TREMOLITET

8equiv.

OF TRBMOLTTE

TABLE 4. SSLECTED INTENATOMIC DISTANCES IN IRE!.IOLITE

t .629 L .627

A(n) -0 (5) 2 .94(7) 2 .95(2)A(B) -0(5) 3 .10(1) 3 .11(2)A(d) -o (6) 2 .78(1) 2 .73(3)A(n) -o (5) 3 .s6(1) 3 .62(3)A(u) -0 (7) 2 .49(2) 2 .49(4)A(e) -0 (7) 2 .58(2) 2 ,63(4)A(b) -0 (7) 3 .79(2) 3 .11(4)A(b) -0 (7) 4 . to (z ) 4 .11(4)

Mean fo r '12

llean for 8 2.84 2.84

A-o(5) 2.973(2) 2.968(L)A-o(6) 3 .148(2) 3 .146(1)A-0(7) 2.480(3) 2,488(2)A-0(7) 3 .680(4) 3 .676(3)

ltean for 12 3.,967- !-992_

1(1) -1 (2) 3 .091( r ) 3 ,095(3)thlough 0(6)

r (1 ) -1 (2) 3 .061(1) 3 .057(3)through 0(5)

r ( r . ) -1 (1) 3 .038(1) 3 .04r . (3 )acroas 0lrror

A(2) -0 (s ) 2 .83(3) 2 .82(5)A(2) -o (s ) 3 .12(3) 3 ,12(s )A(2) -0 (6) 3 .03(2) 3 .02(4)A(2) -0 (6) 3 .27(2) 3 ,27(5 'A(2) -0 (7) 2 .486(4) 2 .495(6 'A(2) -o (7) 3 .684(3) 3 .681(4)

r.607 (2L.637 (2)r.637 (2)r.628(2)

) -0(2))-o (4))-0(5)

)-0 (5)) -0 (6 )

) -].r(1)) -!{(2))-!,r(3)

) -0 (2)4) -0 (4 )

fo! 12 !!qZL !:qS_

for 8 2.867 2.464

* In Tables 2 and 3, the upperX-!ey data, the louer set of

set o f f lgures are fo ! the

f lgu les fo r the neut rod da ta .

IABIJ 3. ANISOTROPIC TMEMTI'RB TACrcR COBFTICIENIS*M (1) -u (4)u (2) -M (3)I 'r(2)-14(4)

-13BtzD33P22{r

0 129(33) 00 105(28) 00(3)

0(2)

n / r \ 1 5 7 ( 1 5 ) 3 4 ( 4 )" ' ^ , r r l ( 1 I ) 3 4 ( 1 )

503(53)436G4)

30(4) 623(56)36(3) 627 (5X)

70 (5) 587 (s6)66(3) 368(43)

58(4) 545(55)56(3) 553(42')

24(6) 1037(88)30(4) 1067(75)

29(2) 361(20)25(4) 263(52)

31(2) 346(19)26(4) 379(55)

39 (1) 397 (36)23(4) 328(61)

30(3) 439(36)34(5) 343(64)

a t A \ T A t r t \ a / t r \

-5(4) 30(17) -9(8)

-10(6) 90(2r) -2(72)-6(4) 74(18) -r9(8)

-32(7) 178(24) -27 (L2)-32(5) 116(18) -14(9)

145G4) 49(4> 494(52'98(10) 44(3' 448(4s)

28L(23) 46(6) 657(77)181(17) 45(4) 598(67)

POLYSBDRAL LMOMS IN MOLITE

Neutrod^ , , , 2 7 3 ( t 6 )" \ " , 2 1 6 ( r o )

^ / < \ 2 1 7 ( r 6 )" ' - l 1 6 0 ( r 1 )

n / 4 \ 2 r 9 ( 1 6 )" . " / 1 5 4 ( 1 0 )

nt1\ 249(24)" . - r 9 3 ( 1 6 )

r / r \ r 5 3 ( 6 )99(14)

r / , \ 1 3 8 ( 6 )' \ - , r 0 0 ( r 4 )

M / 1 \ 1 9 8 ( 1 1 )" ' - ' r 3 0 ( 1 6 )

i l / r \ l 9 r ( 1 1 )" - , r 0 0 ( 1 6 )

M . 1 \ 2 0 8 ( 1 5 )" ' - . 139(22)

i l t , . \ 273(7)" ' - , 248(20)

\ zsi<sz't

0(7) 702(24)-6(5) 75(17)

74(13)76(9)

13(7) 140(23) -71(L2)3 (5) 90 (17) -s8 (10)

0 146 (36) 00 8 8 ( 2 7 ) 0

-s(2) 87(8) -9(4)2(6) 47 (22) -8(10)

-5 (2) 78 (8) -6 (4)-5 (6) 47 (24) -110.0)

0 1r.3(15) o0 64(26) o

0 r.00(r.5) 0o 64(26) 0

o 67 (22) 00 62(36) o

T(l) Tetrehedron

o(1) -o (s ) 2 .6s1Q)8, 2 .6s6Q)80(1)-0(6) 2.682(2) 2.680(2)o(1) -o (7) 2 .67J" (3) 2 .664(2)0(5) -0 (6) 2 .618(3) 2 .620(2)0(5) -0 (7) 2 .632(3) 2 .$2O)o(6) -o (7) 2 .652(2) 2 .644(2)

2.tt2- 24?6-u(L) ocrahedrotr

2,81-7 (2) 2.877 (2)3 . 0 6 7 ( 2 ) 3 . 0 7 2 ( 2 )2 . 7 4 7 ( 3 ) 2 . 7 5 6 ( 2 )3 . 0 5 7 ( 3 ) 3 . 0 5 6 ( 2 )2 . 8 8 4 ( 3 ) 2 . 8 8 4 ( 3 )3 . 0 8 1 ( 2 ) 3 . 0 8 2 ( 1 )2 . 6 8 s ( 5 ) 2 . 6 9 7 G )

M(2) octahedrotr

2 . 7 s 2 ( 3 ) 2 . 7 5 3 < 2 )2.8L7 (2) 2.8r7 Q)3 . 0 4 3 ( 2 ) 3 . 0 4 0 ( 2 )3 . 0 0 7 ( 2 ) 3 . 0 1 0 ( 2 )3 . 0 0 5 ( 2 ) 3 . 0 0 5 ( 2 )2.909(2) 2.9o6(2t2.992(4) 2.999(3)

2 . 9 4 2 2 . 9 4 2

0 r1 - ) -o a2- )o ir l i -o izj !0 ( 1 : ) - o ( 3 : )0 ( 1 - ) - o ( 3 - )o ( 2 ) - o ( 2 )0 ( 2 ) - 0 ( 3 )0 ( 3 ) - 0 ( 3 )

Mean

28(4')r6 (6)

40(2)44 (5)

432(52)319(84)

783 (23)873(81)

0 ( 1 ) - 0 ( 1 ) r0 ( 1 - ) - 0 ( 2 : )0 (1-) -0 (2-)0 ( 1 ) - 0 ( 4 ) ro ( 2 : ) - o ( 4 : )o (2-) -o (4-)0 ( 4 ) - 0 ( 4 )

Mean

0 302(10) o0 340(32) o

0 5(90) 0

M(1) -0(1) 2 .06L(2) 2 ,063(L)M(1) -0(2) 2 .07r (2 ) 2 .o72(2)M(1) -0(3) 2 .082(2) 2 .087(2)

ueatr 2.9]]- 2.-07_t_

M(2) -0( r . ) 2 . r43(2) 2 ,146(2)M(2) -0(2) 2 .09r (2 ) 2 .087(1)M(2) -0(4) 2 .0 r8Q) 2 .078(2)

1(2) Tetrahediotr

o(2)-o(4) 2.142(pr8 2.74oQ>8.0 ( 2 ) - 0 ( 5 ) 2 . 6 7 s ( 2 ' 2 . 6 7 5 ( 2 )o ( 2 ) - o ( 6 ) 2 . 6 6 3 ( 2 > 2 . 6 7 0 ( 2 )o(4)-o(5) 2.656(3) 2.656(2>o(4)-o(6) 2.562(2) 2.566(2)o(5)-o(6) 2.694G) 2.697(2)

Men 2:99t_ 2.,567_

u(3) octahedrotr

0 1 r " 1 - o 1 r l y z . ; s z 1 : ; 2 . 7 5 3 ( 2 ,o ( r : ) - o ( u ) 3 . 0 9 i ( 3 ) 3 . 0 9 3 ( 2 )o ( 1 : ) - . o ( 3 : ) 2 . 7 4 7 ( 3 ) 2 . 7 s 6 ( ? '0 ( 1 - ) - 0 ( 3 - ) 3 . 0 6 9 ( 3 ) 3 . 0 6 8 ( 2 )

kan

u(4) Adti-pl1s

o ( 2 ) - o ( 2 ) . . 2 . 8 8 4 ( 3 ) 2 . 8 8 4 ( 3 )o ( 2 - ) - 0 ( 4 ; ) 3 . 1 4 4 ( 2 ) 3 . r 4 7 ( 2 )o ( 2 : ) - 0 ( 4 : ) 3 . o o s ( 2 ) 3 . o 0 5 ( 2 )o ( 2 : ) - o ( 5 ; ) 3 . 6 3 8 ( 3 ) 3 . 6 4 0 ( 2 'o ( 4 : ) - o ( 5 : ) 3 . 4 4 4 ( 3 ) 3 . 4 4 8 ( 2 'o ( 4 : ) - 0 ( 6 : ) 2 . 5 6 2 ( 2 ) 2 . 5 6 6 ( 2 to ( s - ) - 0 ( 6 ; ) 2 . 6 1 8 ( 3 ) 2 . 6 2 0 ( 3 >0 ( 5 " ) - 0 ( 6 - ) 3 . 0 8 2 ( 3 ) 3 . 0 9 0 ( 3 )0 ( 6 ) - 0 ( 6 ) 3 . 4 5 7 ( 3 ) 3 . 4 5 s ( 3 )

Meen 3.084 3.086

ut, " ao

218(18) 2455(236)

338 THE CANADIAN MINERALOGIST

of ferrotschermakite (Hawthorne & Grundy1973a), refinement of all variables for an iso-tropic thermal model resulted in convergenceat an R-factor of.3.2Vo. At this stage, tempera-ture factors (except those of the split .,4-site)were conyerted to anisotropic and a sorregtionfor isotropic extinction (Zachariasen 1968) wasintroduced into the refinement. Several cyclesof least-squares refinement gradually increasingthe number of variables resulted in convergenceat an R-factor of Z.lVo. There is a substantialdifference between the scattering amplitudes ofaluminium and silicon, and thus tle site-occu-pancies of the tetrahcdral sites were consideredas variable with the 6u11 gfiemistry of the siteconstrained to be the s4me as that given by thechemical analysis. The following site-occu-pancies were obtained: ?(1) =0.06(4)Alf 0.96Si;T(2):0.00A191.00Si; in view of the fact thatthe Al occupancy of T(1) is within 2c of zero,this result cannot be considered as significant.

X-ray

Analytic scattering curves for neutral atomswere taken from Cromer & Mann (1968) withanomalous dispersion coefficients from Cromer& Lieberman (1970). Proceediqg as for theneutron refinement, one cycle of least-squares

varying the positional parameters gave an R-factor of 6.0Vo , and a further cycle varying theisotropic temperature factors resulted in an R-factor of 5.6Vo. An additional cycle varying allparameters for an isotropic thermal model re-sulted in an R-factor of. 5,2Vo. This was firrtherreduced to 4,8Vo by positionally disordering thez4-site and further refinement. At tlis stage,temperature factors were convert"U 1o aniso-tropic and a variable extinction correction wasintroduced. The hydrogen parameters from theneutron refinement were inserted at this stage,but were not allowed to vary. Several cycles ofleast-squares refinement gradually increasingthe number of variables resulted in convergenceat an R-factor of 3.'l,Vo*.

Final atomic positions and equivalent iso-tropic temperafure factors are presented inTable 2 and anisotropic temperature factors inTable 3. Interatomic distances and angles, andthe magnitudes and orientations of the principalaxes of the thermal ellipsoids, were calculated

*Tables of structure factors can be obtained from:the Depository of Unpublished Data, NationatScience Library, National Research Council of Can-ada, Ottawa, Ontario, Canada.

?ABLE 6. SELECTED INIEIATOMIC ANGLES IN TREMOLITE

Neutrotr X-Ray Neutro!

o(1)-r(1)-0(5)o(1)-r(r)-0 (6)0(1)-r ( r ) -0 (7)0 (s)-1(1)-0(6)o(5)-r (1)-0(7)0 (6)-r (1) -0(7)

!,lean

0 (lu) -u(1)-0(2u)0 (lu)-!r(1)-o(2d)o(1u)-!1(1)-0 (31)0 (lu)-!t(1)-0 (30)0 (2)-11(r.)-0(2)0 (2)-u(1) -0(3)0 (3) -!{(r)-0(3)

Ueao

o(1)-r(2)-0(L)0(ru)-!r(2) -0 (?u)0 (r)-u(2)-0(2d)0 (1)-u(2)-0(4)0 ( 2u) -r.{(2)-0 (4:)o (2u)-M(2) -o (4q)0 (4)-lr(2) -0 (4)

Uean

o(7) -0(7)-o(7)

0 (3t ) -0(3)-n

r .12 .4 ( r ) '

111 .4 (2 )110.9 ( r )106.3 (1 )r07 .5(1)108.2(2)

lEar_

96.0 (1 )8 5 , 9 ( 1 )94 .8 ( r )83 .2 (L )88 .2(1)9 5 . 7 ( L )8 0 . 5 ( 1 )

9 0 . 0

7 9 . 8 ( 1 )e 1 . 8 ( 1 )8 3 , 4 ( L )92 .5(1)90 .1 (1 )94 . r . (1 )95.9 (2)

6 5 . 9 ( 1 )o.268

90.4(2J

0 (2) -r (2)-0 (4)o (2) -r(2)-0 (s)0 (2)-r (2)-0(6)0 (4) -r(2)-0 (5)0 (4)-r (2) -0(6)0 (5)-r (2) -0 (6)

Itean

o (ru) -lr(3) -0 (r:)0 (lu) -u(3) -0 (10)o (lu) -u(3) -0 (3:)0 (lu) -u(3) -0(3d)

l.Iean

o ( 2 ) - M ( 4 ) - o ( 2 ) .0 (2u) -M (4 ) -0 (4q)o (2u) -!1(4)-o (4u)o (2u) -M(4)-o (5:)0 (4u) -M(4) -0 (50)0(4u)-tr(4) -0 (6Y)0 (5u) -lr(4):0(60)0 (su)-M(4) -0(6u)o (6)-l i(4) -o (6)

Mean

r (1) -0 (5 ) - r (2 )r(1) -0 (6) -r (2)r(1)-0 (7) -r(1)0 (5) -0 (6) -0(5)0 (5 ) -0 (7 ) -0 (6 )

T(l) Tetrahedlon

112.3 (1 ) .

111.2(1)r11 .1(1)106. t,(1)107.4 (1)108.6 (1 )

!!9:-1-M(l) oetahedlotr

9 5 . 8 ( 1 )8 5 . 9 ( 1 )9 5 . 1 ( r )83 . l " (1 )88 .2( r )9s,8( r )80.3 (1)

g9:g_-

It(2) 0ctahedron

79.9(L)91 .9 (1 )83 .4 ( r )92 .5 (1 )90 .1 (1 )94 .0(1)95 .5(2)

9 0 . 0

$ Polvhedron*

65.8 ( r )

!:3u'

T(2) letlabedron

r17 .8( r ) " 117.6(1) .109.5(1) rO9,5(1)108.2(1) r08 .1(x )109.6(1) 109.9(2)r03 .4(1) 103.5(1)107;9(1) 107.7(1)

109=l- !!9il_

U(3) octshedton

96.8(2) 96 .7 (2 )83 .2(2) 83 .3(2)96 ,2(L) 96 .1(1)83 .8(1) 83 .9(1)

9!.-o 9.9:A-u(4) Antt-prl.si

7 3 , 1 ( 2 ) 7 3 . 1 ( 2 )7 8 . 6 ( 1 ) 7 8 . 5 ( 1 )8 2 . 9 ( l ) 8 3 . 0 ( 1 )8 9 . 0 ( 2 ) 8 9 . 0 ( 2 )8 4 . 5 ( 1 ) 8 4 . 5 ( 1 )63 ,0(1) 63 . r . ( r )7 1 . 0 ( 1 ) 7 0 . 9 ( 1 )5 8 . 8 ( 1 ) 5 8 . 8 ( 1 )8 5 . 5 ( 1 ) 8 5 . 2 ( 1 )

1L.e 1l-9:Tetrahedlal Chal.u

136.2(r) 136.1(1)137.8( t ) L37.7(L)r37,8(2) t38.2(2)167.6(1) 167.6(1)166,4(L) 166.6(1)

* A = ( 90 ' - 4 . o (7 ) -o (1 ) -0 (7 ) ) / 90 . .

CRYSTAL STRUCTIJRB OF TRBMOLITE

TABLE 7. ELLI?SOIDS OF VIBRATION POR TRNTOLITE

339

R.U.S. Dlaplac@ent

a2A!g1e to a-qis ln AngLe to b-sis l! Angle to c-qla tu

degreea degrees deglees

X-&ay NeutroD X-Ray Neutro! X-Ray Neutron X-Ray Neutlotr

0.075(4) 0 .068( i )0 ( r ) 0 . 0 8 r ( 4 ) 0 . 0 7 7 ( 3 )

0 .088(4) 0 .080(4)

0 . 0 7 5 ( 4 ) 0 . 0 6 5 ( 4 )0(2) 0 .082(4) 0 .074(4)

o .o92(4) 0 .088(3)

0 . 0 8 7 ( 5 ) 0 . 0 8 6 ( 8 )0 ( 3 ) 0 . 0 9 2 ( 5 ) 0 . 0 8 6 ( 6 )

0 . 1 r 3 ( 5 ) 0 . 0 9 3 ( 4 )

0 .070(5) 0 .066(3)0(4) 0 .084(4) 0 .084(4)

n 1 2 1 l a \ n r n o / r \

0.074(5) 0 .063(4)0(5) 0 .100(4) 0 .086(3)

0 . u7 (3 ) o . u .4 (2 )

0 . 0 6 1 ( 5 ) o . o 7 r ( 4 )0 ( 6 ) 0 . 1 0 r ( 4 ) 0 . 0 8 5 ( 3 )

0 . u 0 ( 3 ) 0 . 1 0 6 ( 3 )

0 . 0 6 3 ( 7 ) 0 . 0 7 r . ( 5 )0(7) 0 .106(5) 0 .094(4)

n 1 r r l ( \ d 1 7 1 l J . \

0 . 0 6 5 ( 2 ) 0 . 0 5 7 ( 6 )r (1 ) 0 .069(2) 0 .065(5)

0 .085(2) 0 .068(5)

0 .065(2) 0 ,062(5)r (2 ) 0 .071(2) 0 .069(5)

0 ,081(2) 0 .073(5)

0 .068(3) 0 .061(5)M(r ) 0 .080(3) 0 .065(7)

0 . 0 9 6 ( 3 ) 0 . 0 7 8 ( 5 )

0 . 0 7 0 ( 3 ) 0 . 0 6 5 ( 7 )M ( 2 ) 0 . 0 7 4 ( 3 ) 0 . 0 7 2 ( 5 )

0 .094(3) 0 .074(5)

0 .068(5) 0 .052(9)u(3) 0 .076(5) 0 .064(9)

0 .099(4) 0 .080(6)

o .072(2) 0 .064(7)M(4) 0 .081(2) 0 .085(5)

0 .123(1) 0 .125(4)

0.108 (r.1)E - - - 0 .185(9)

0 . 1 8 9 ( 8 )

94 (r5) 47 (r5)? o / r e \ r l o / r 1 \

L2(27) L24(28)

L44(2r) 10(r.4)Lr4(27) 96(19)

64( r .3 ) 83(6)

90 132 (36)77 (10) 90r.3(r.0) 41(31)

63(6) 57(4)7o(8) 68(7)35(4) 4 r . (4 )

102(9) 108(7)166(8) 158(6)97 (9) r.o2 (4)

1 1 / / < \ r r < / 1 n \

r48(12) r53(10)109 (16) 98 (6 )

90 90L77 (2O) 165 (7)87(20) 10s(7)

106(8) 105(23)69(8) 105(69)27 (4) 21(51)

LL2(7) 67 (31)109 (9) 156(3r.)

30(6) 86(67)

[1 (5) 9090 73(20)21 (5 ) 17 (20)

o n r r < 1 ? Q \

7r. (6) 145 (38)19(6) 90

7 (23) 64(17)95 (28) 29(26)85(15) 103(37)

107 (r.5) 87 (e)rL2(2o) 1170.0)r.52 (r.6) r52 (i"o)

27 (6) 34(6)100(13) 113(7)115 (3 ) r14 (3 )

121 (4) r.19 (3)90 (9 ) 91 (5 )31(3) 29(3)

56 (4 ) 56 (4 )8 6 ( 1 5 ) 9 8 ( 7 )

146(4) 145(4)

63(20) 65(21)30(19) 33(51)

102(5) 70(64)

84 (14) 35(24)r.61(8) 7L(47)108(8) 1 r8(34)

95(28) 66(13)174(28) 88(39)

94Qe) 24(13')

4s(29) 114(18)135(28) 146(14)

86( r .5 ) 66(10)

90 21 (36)179 (10) 90

91( r "0 ) 63(31)

96(14) 106(8)168(8) 156(7)

80(5) 72(6)

31(4) 30(3)90 (8 ) 97 (6 )60(4) 61(3)

3 7 ( 4 ) 3 7 ( 5 )1 ^ r / r 1 \ r n 1 / o \s9 (7 ) s5 (4 )

90 9012(20) 9 r . (7 )18(20) 1 (7)

27 (2O) 25(27)116(21) 114(30)

8r (4) 9'1 (36)

9(7) 73(23)8s(15) 62(68)81(s) 34(62)

6(5) 9090 L78(20)84(5) 88(20)

90 20 (38)776(6) 110(38)86(6) 90

90 90r .68(8) 178(23)102(8) 92(23)

3r (1 ) 38(4)90 9059(1) 52(4)

89 (6 )119 (6)

90

0 9 09 0 090 90

0 090 9090 90

90 9087(8) 78(23)3(8) r2Q3')

900

90

090

9090

09 0

90900

09090

900

90

9090

0

136(1) 143(4)90 9046(1) 53(4)

r .6 (6)7 4 ( 6 )90

with the program ERRORS (L. W. Finger, pers.comm.)r and are presented in Tables 4-7.

DrscussloN

The general topology o1 fts amphiboles hasbeen discussed in detail elsewhere (e.g. Papike& Clark 1968; Papike & Ross 1970) and willnot be repeated here. Figure 1 illustrates theC2/m ampfubole structure and may be used inthe interpretation of Tables 2-7. A comparisonof the mean metal-oxygen distances obtainedfrom thh study with those obtained by Papikeet al. (1,969) is given in Table 8. The [T(l)-Oldistance is significantly larger in this studn in-

dicating that the small amount of tetrahedralaluminium present is strongly ordered into the7(1) site in accord with previous studies (Papike& Clark 1967; Robinson e, aI. 1973; Haw-thorne & Grundy 1973a,b,1976).T:he tM(l)-Oland [M(3)-O] distances are slightly less in thisstudy due to the partial occupancy of O(3) byfluorine. Using the regression equations ofHawthorne (in prep.) mean bond lengths of2.o97_Q)A and 2.060(7)A are forecas! for M(1)and MQ), in good agreement with the experi-mental values. The tMQ)-Ol distance is slightlylarger than that observed by Papike et al., (1969);this arises from the small amount of octahedral

340 TIiE CANADIAN MINERALOGIST

TABLE 8. MEAN UETAL-OXYGEN DISTANCES FOR TREMOLITE.

r(r.)-0 r(2)-0 u(r")-0 M(2)-0 l,r(3)-0

Paplke af, aL. (L969) 1.6201 1.6321 2.075l. 2.077L 2.0668

X-!ay (this scudy) L.629 1.635 2.071 2.084 2.063

Neut ron ( th ls s tudy) 1 .627 1 .635 2 .074 2 .084 2 .065

aluminirrm that occurs in the hydroxy-tremoliteand which presumably orders into the MQ) srte.For complete MQ) occupancy by magnesium,the regression equations of Hawthorne (in prep)forecast a mean M(2) bond length of 2.084(5)A,in exact agreement with the value obtained inthis study.

It is well known that anisotropic thermalparameters derived from X-ray data can absorbdeviations from spherical symmetry of theatomic charge cloud (Coppens L968), and thustemperature factors obtained by X-ray crystal-structure refinement using spherical atomic formfactors are systematically in error because ofbonding effects. This effect is not present ina neutron crystal-strucfure refinement, and thederived anisotropic thermal parameters closelyrepresent the vibrational behavior of the atoms

provided anharmonic effects are negligible.Thus a comparison of the results obtained fromeach experiment should glve an indication ofthe systematic error in the derived X-ray struc-ture model. In principal, these effects could alsoaffect the atomic positions; however, this ap-pears to be negligible for all except the lightestatoms of the first row of the periodic table.Half-normal probability plot analysis of the twosets of results shows that, whereas the atomicpositions correspond very closely, signifisanlsystematic differences occur between the aniso-tropic temperature factors.

Delocalization of electron density will causean increase in the temperature factors obtainedfrom the X-ray refinement and examinafiea sfTable 2 shows that, in general, the X-rayequivalent isotropic temperature factors arelarger by l0 to 5OVo than the correspondingneutron values. The two exceptions to this arethe M(4) and O(7) sites where the equivalentisotropic temperature factors are equal; thismay result from positional disorder of the Caand Na on the M(4) site and a reaction of O(7)to positional disorder on the I site.

A comparison of the magnitudes of the prin-

L-'l"o

mm n m

Ftc. 1. The C2,/m clino-amphibole structure projected down a* (after Hawthorne & Grundy 1973a).

CRYSTAL STRUCTURE OF TREMOLITE

o. to

o .o8

o.o6

0 .06 0 .o8 o. ro o . l2

NEUTRON VALUES tA I . - - - - - - 'Frc. 2. Comparison of the magnitudes of the principal axes of the tiermal

ellipsoids of the anions in tremolite from the X-ray and neutron refine-ments.

o. to

o.o8

TETRAHEDRAL

CATIONS

OCTAHEDRAL

CATIONS

. M(4) SITE

o.o6

o .o6 o .o8 o . t2

N E U T R O N V A L U E S I A I - - *Fro. 3. Comparison of the magn.itudes of the principal axes of the thermal ellipsoids

of the cations in tremolite from the X-ray and neutron refinements.

341

I5at dfJ

t' Ix

A

II

€alrJlJ

G.Ix

N O N _ B R I D G I N G

A N I O N S

B R I D G I N G

A N I O N S

342 TI{E CANADIAN MINERALOGIST

Ftc. 4. Comparison of the orientations of the principal vibration axes of the anions io tremolite derivedfrom X-ray and neutron data

cipal axes of the thermal ellipsoids (Figs. 2, 3)shows that the X-ray values generally exceedthe neutron values as expected. The amount ofanisotropy is generally less for the seutron re-sults, but it is apparent from both studies thatthis is affected by the coordination number ofthe atom (Figs. 2, 3). For the anions, the meandifference between the magnitude of the max-imum and minimum principal vibration axeschanges from 0.054A (X-ray) and 0.050A(neutron) for two-fold coordination, to 0.048Aand 0.032A for three-fold coordination. to0.019A and 0.014A for four-fold coordinaiion.$imilafly, values for the cations change from0.0184 and 0.011A for four-fold coordination,to 0.028A and 0.017A for six-fold coordina-tion, 0.051A and 0.062A for eight-fold coordina-tion.

The orientations of the thermal ellipsoid prin-cipal axes are compaled in Figures 4 and 5.The principal axes of O(1) are only slightly

anisotropic. The neutron values approximate aslightly oblate spheroid with X as the charac-teristic axis; this axis is sub-parallel to ?(1)-O(1), the strongest bond to the O(1) anion. TheX-ray vibration axes do not correspond closelyto the neutron values; however, the X-ray valuesare not well-defined as the amount of aniso-tropy is small. For the O(2) anion, both studiesindicate a slightly prolate spheroid with Z asthe characteristic axis. The minimum vibrationdireition (neutron) is sub-parallel to T(2)4(2),the strongest bond to the O(2) anion. The corre-spondence between the axis orientations in bothstudies is quite close. For O(3), the neutronresults indicate an almost isotropic behaviorwhereas the X-ray results show strong anisotropythat approximates a prolate spheroid wrth Z asthc characteristic axis. Examination of Figure 4shows tlat the Z axis is parallel to the O(3)-Hbond, and it is apparent that delocalization ofelectron-density along this bond has significantly

x Iztn(ll + ^r{l I"f=o

I

rPlo \

trt.J ZtzY

CRYSTAL STRUCTURE OF TREMOLITE 343

Ftc.4. See caption on page opllosite.

affected the temperature factor values in tleX-ray refinement. For the O(4) anion, bothstudies indicate that the principal D(es corre-spond to triaxial ellipsoids whose orientationsare statistically identical.

As with O(1) and O(2), the minimum prin-cipal vibration direction is subparallel to thebond formed with the coordinating tetrahedralcation. For the chain-bridging anions O(5),0(6) and O(7), both studies indicate significantanisotropic behavior. All principal axes fromthe neutron study are triaxial ellipsoids where-as, from tle X-ray study, O(5) is a triaxialellipsoid but 0(6) and O(7) are approximateoblate spheroids. For these atoms, identicalorientations of the thermal ellipsoids were ob-tained from both studies (Fig. 4). The minimumvibration directions are always sub-parallel tothe tetrahedral bonds whereas the intermediateand maximum principal vibration directions areapproximately perpendicular to tle tetrahedralbonds in all cases. This is as expected since the

r I-IAY ELUPIO|D AxEs lI,Y.Zl. NEUTION EUJPSOIO AXESIXYZI

- ONE STANOAnO DEVIATIONIUPPEI HEnl t?HElElI.. ONE STANDARD OEVIATIONITo\TEl XCn|S'HETEI. COORDINATING ATOmSIUPPET ilCnlS?HElClo cootDtNATtNG AtOmS lro$tr xttgflEtEl

bond-stretching moments are considerably largerthan the bond-bending moments, and it agreeswith the infrared spectrum (unpublished data)which shows strong Si-O-Si bending modes. Theneutron results for ttre tetrahedral sites T(1)and T(2) (Fig. 5), show their vibration to bestatistically isotropic. The X-ray results showa slight anisotropic behavior, the principal vibra-tion axes approximating oblate spheroids withX as the characteristic axis. For the octahedralsites, the neutron results indicate that the MQ)vibration is essentially isotropic whereas theM(l) and M(3) vibrations are slightly aniso'tropic with the principal vibration a.xes colre-sponding to prolate spheroids with Z as thecharacteristic axis. The principal vibration axesfor M(l) and M(3) are normal to the octahedralfaces and hence the maximum vibration involvesthe minimum possible bond-stretching. For theM(1) site, the X-ray results indicate principalvibration aJ(es corresponding to a triaxial ellip-soid with essentially the same orientation as that

THE CANADIAN MINERALOGIST

obtained in the neutron study (the reversal ofthe X and Y axes is not significant due to thespheroidal nature of the neutron vibration ellip-soid). From the X-ray results, the MQ) andM(3) vibration ellipsoids approximate prolate

spheroids with Z as tle characteristic axis. Bothstudies show that the M(4) vibration is stronglyanisotropic with the principal vibration axesapproximating a prolate spheroid w|rth Z as thecharacteristic axis. The orientation of the prin-

sr,L ^ogr s,F z

/ \

016l

^ Znrir 1x+

I

olrl

- o

o(21

I

2 7 o 0 )- a

oizt otozt

./v , /

o(t)o

z z o(31|{{f+{ a

o(uo

, - /

o(ua

0ltlo

qua

a

o{6}

qslz o

a

0(51

0{61

I

{

r X

Frc. 5. Comparison of tle orientations of the principal vibration axes of the cations in tremolite derivedfrom X-ray and neutron data. Legend as for Figure 4.

CRYSTAL STRUCTURE OF TREMOLITE 34s

cipal axes is the same in both studies; theminirriusl vibration direction is sub-parallel toM(4)4(4), the strongest bond, whereas themaximum vibration is perpendicular to thisbond and bisects the angle between the M(4)-O(2) and M(4)-O(5) bonds.

It is apparent that the disposition of bondsaround an atom strongly controls both thedegree of anisotropy in its vibration and theorientation of the principal vibration directions.The minimum vibration direction tends to beoriented along the strongest bond with themaximum vibration direction bisecting the anglebetween the weaker bonds. As the coordinationbecomes more regular, this tendency decreasesand the vibrations become more isotropic.

The magnitudes of the principal axes of thethermal ellipsoids are systernatically larger forthe X-ray refinement, indicating some contribu-tion from bonding and ionization effects. Theorientations of the principal axes of the thermalellipsoids can be related to the strength andorientation of the bonds to the coordinatingatoms. The minimum vibration direction isgenerally sub-parallel to the strongest bond;this becomes less apparent as the coordinationand the strengths of the bonds become moreregular.

The present neutron study confirms the hy-drogen position in tremolite found by Papikeet al. (1,969). The O(3)-H bond length of0.960(6)A is fairly typical for a hydroxyl andshows'that little or no hydrogen bonding withthe chain-bridging anions occurs. This is sup-ported by the infrared spectrum which shows asingle narrow transition at 3684cm{. This valueagrees well with the relationship proposed byHamilton & Ibers (1968) relating principal OHstretching frequency to nearest-neighbor aniondistance. The vibration ellipsoid of the hydrogenis an oblate spheroid with its tbree principal axesparallel to a*, b and c. Because of the site-sym-metry of the hydrogen position (rn), one prttcipal axis is constrained to be parallel to theD-axis and the other two to lie in the mirrorplane. The direction of minimum vibration isparallel to a'! and the O(3)-H bond. Thisorientation, together with the significant aniso-tropy of the hydrogen vibration, is to be ex-pected as bond-stretching moments are muchlarger than bond-bending moments.

AcrNowl-sDGMENTs

The authors would like to thank Dr. MartinAnderson for his help in this work. Financialsupport was provided by the National ResearchCouncil of Canada (grant to H.D.G.) and theOntario Government (fellowship to F.C.H.).

REFERENCES

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CoppE\rs, P. (1974): Some implications of combinedX-ray and neutron diffraction studies. Acta Cryst.B,30,255-261,

CRoMER, D. T. & Mexx, J. B. (1968): X-ray scat-tering factors computed from numerical Hartree-Focf wave functions. Acta Cryst. A24,321'324'

& LTgSERMAN' D. (1970): Relativisticcalculation of anomalous scattering factors forX-rays. J, Chem. Phys. 53, 1891-1898.

FrNcBn, L. W. (1969a): RFINE. A Fortran fV com'puter program for Sructure factor calcuation andleast-squares refinement of crystal. struc-tue$.Geophys. Lab. Carnegie lttst. IVash. (unpublishedmanuscript).

HAMTLToN, W. C. & Inms, J. A. (1968): HydrogenBonding in Solids. Benjamin, N.Y.

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& - (1973a): The crystal chem-istry of the amphiboles. I. Refinement of thecryital structure of ferrotschermakite. Mineral-Mas. 39,3648,

& _ (1973b): The crystal chem-istry of the amphiboles. II. Refinement of thecryital structure of oxykaersotite' Minetal. Mag.39, 390-400.

& -- (submitted for publication):The crystal chemistry of the amphiboles. III.Refinement of the crystal structure of a sub-silicic hastingsite.

PAprKE, J. J. & Cr.enrc, J. R. (1967): Crystal-chem-ical role of potassium and aluminium in a horn-blende of proposed mantle origin. Geol, Soc.Amer. Spec. Pap. 11^5, 171 (1968) (Abstr.).

& - (1968): The crystal strucrureand cation distribution of glaucophaue. Amer.Mineral. 63, 1156-1173,

& Ross, M. (1970): Gedrites. Crystalstrucrures and intracrystalline cation distributions.Anier. Mlneral. 55, L945-L972.

& Cr-enr, J. R. (1969): CrYs'tal-chomical characterization of clinoamphibolesbased on five new structure refinements. Mineral-Soc, Amer, Spec. Pap.2, 117-136,

RoBB.{soN, K., Grnng G. V., Rrnrq P. H. & Her-r-,M. R. (1973): Catiou distribution in three horn-blendes. Amer. t. Sci. 27XA, 522-535.

Wennnr, B. E. (1929): The structure of tremolite"HgCaaMgs(SiaOrJ. Z. Krist. 72, 42'57.

ZAcEARIAsEI.I, W. H. (1968): Experimental tests ofthe general formula for the integrated intensityof a real crystal. Acta Cryst. L24,212-2t6.

ZussMAN, J. (1959): A re-examination of the struc'ture of tremolite. Aaa Cryst. t2,3O9-3t2,

Manuscript recelved March 1976, emended April1976.